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∫calculus i review

key term - Inverse hyperbolic functions

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Definition

Inverse hyperbolic functions are the inverses of the hyperbolic functions, such as sinh, cosh, and tanh. They are used to solve equations involving hyperbolic functions.

5 Must Know Facts For Your Next Test

  1. The inverse hyperbolic sine function is denoted as $\sinh^{-1}(x)$ or $\text{arsinh}(x)$ and is defined by $\sinh^{-1}(x) = \ln(x + \sqrt{x^2 + 1})$.
  2. The inverse hyperbolic cosine function is denoted as $\cosh^{-1}(x)$ or $\text{arcosh}(x)$ and is defined only for $x \geq 1$ by $\cosh^{-1}(x) = \ln(x + \sqrt{x^2 - 1})$.
  3. The inverse hyperbolic tangent function is denoted as $\tanh^{-1}(x)$ or $\text{artanh}(x)$ and is defined for $-1 < x < 1$ by $\tanh^{-1}(x) = \frac{1}{2}\ln(\frac{1+x}{1-x})$.
  4. Inverse hyperbolic functions can be derived from their corresponding exponential definitions and logarithmic identities.
  5. They have applications in various fields including calculus, complex analysis, and physics.

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